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First Order Expansion in the Semiclassical Limit of the Levy-Lieb Functional
Maria Colombo1, Simone Di Marino2, Federico Stra1,3
1EPFL, AMCV, Lausanne, Switzerland.
Archive for Rational Mechanics and Analysis
|November 10, 2025
Summary
This study proves the first-order expansion of the Levy-Lieb functional in the semiclassical limit of Density Functional Theory (DFT). Researchers established bounds for zero-point energy in systems with two electrons, using optimal transport and Dirichlet penalization.
Area of Science:
- Quantum Chemistry
- Mathematical Physics
- Computational Physics
Background:
- Density Functional Theory (DFT) relies on approximations for electron interactions.
- The Levy-Lieb functional is crucial for accurate DFT calculations, especially in the semiclassical limit.
- Understanding electron behavior at the quantum level requires precise functional approximations.
Purpose of the Study:
- To rigorously prove the conjectured first-order expansion of the Levy-Lieb functional.
- To establish asymptotic lower and upper bounds for this functional in specific physical systems.
- To connect DFT approximations with fundamental quantum mechanical principles.
Main Methods:
- The study employs techniques from singular perturbation theory.
- Optimal Transport theory is utilized to model electron interactions.
- Dirichlet penalization is applied to analyze the functional's behavior.
Main Results:
- A general asymptotic first-order lower bound for the Levy-Lieb functional is proven.
- An asymptotic upper bound is derived for the specific case of two electrons in one dimension.
- The results validate theoretical predictions in the semiclassical regime.
Conclusions:
- The findings provide a rigorous mathematical foundation for DFT approximations.
- This work advances the understanding of electron correlation and kinetic energy functionals.
- The methods offer a new perspective on solving complex quantum many-body problems.
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